334) Mr. Graves's Explanation of a remarkable Paradox 



Gravitation = - „, 



thai is, the attractive force varies inversely as the square of 

 the distance. 



By substituting for p its value a [Y—e-) we get the ex- 

 pression usually given in books on physical astronomy. 



It may be proper to remark that the foregoing proof dif- 

 fers but little in principle from one lately given by M. Tran- 

 son, in the Jotirfial de Mathematiques ; but as that is made to 

 depend upon a property involved in the theory of caustics — a 

 subject quite foreign to the inquiry— it is presumed that the 

 process here given will be preferred on the score of elementary 

 simplicity. 



Belfast, Sept. 26, 1836. 



LXVI. Explanation of a remarkable Paradox in the Caladus 

 of Functions, noticed by Mr. Babbage. By John T. 

 Graves, JEsg., M.A., of the hmer Temple.^ 



nPHE following passage occurs in a very valuable paper by 

 -*- Mr. Babbage, " On the Analogy between the Calculus of 

 Functions, and other branches of Analysis," published in the 

 Philosophical Transactions for 1817. (vol. cvii. pp. 21 1, 212.) 



" Here perhaps it may not be misplaced to state a diffi- 

 culty of a peculiar nature with respect to functional equations 

 which are impossible ; for the sake of perspicuity I shall con- 

 sider a very simple case, 



■if X =■ cif — (1.) 



for X substitute — , ^ — = C'i^x (2.) 



XX 



and by multiplication, J/ a' x4/ — =■ (? ■^ x -^ ■if — , or 1 =c-, 



from which it follows that c = + 1, or, in other words that 



the equation 4/a" = c\J/ is contradictory, unless c = + 1. 



Now the functional equation F {^, rj/ ^, \I/ a .r} =0 has been 

 reduced by Laplace by means of a very elegant artifice to an 

 equation of finite differences ; nor am I aware that this pro- 

 found analyst has pointed out any restriction or any impossi- 

 ble case. If we treat the equation vj/ x = cil;.r" by his me- 

 thod, we shall find for its solution 



— log log a s 



ifx = c '°s» (3^j 



• Communicated by the Author. 



